Bücher der Reihe Cambridge Tracts in Mathematics

'Martin's axiom' is one of the most fruitful axioms which have been devised to show that certain properties are insoluble in standard set theory. It has important applications m set theory, infinitary combinatorics, general topology, measure theory, functional analysis and group theory.

The aim of this book is to promote a fibrewise perspective, particularly in topology. Already this view is standard in the theory of fibre bundles and therefore in such subjects as global analysis. It has a role to play also in general and equivariant topology.

Heat Kernels and Spectral Theory investigates the theory of second-order elliptic operators.

The second edition of this celebrated book contains a full and self-contained proof of De Branges' theorem. Every chapter has been updated and many of the original proofs have been simplified. It will be essential for all interested in complex function theory.

The new edition of this well known book remains a clear and accessible exposition of the basic theory of Hardy spaces. The coverage has been extended with the inclusion of two appendices; in addition numerous amendments, additions and corrections have been made throughout.

The purpose of this book is to present the theory of general irreducible Markov chains and to point out the connection between this and the Perron-Frobenius theory of nonnegative operators. The author begins by providing some basic material designed to make the book self-contained, yet his principal aim throughout is to emphasize recent developments.

The theory of asymptotic expansions is of great importance in many branches of pure and applied mathematics and in theoretical physics. Solutions of ordinary differential equations are frequently obtained in the form of a definite integral or contour integral, and this tract is concerned with the asymptotic representation of a function of a real or complex variable defined in this way.

The theory of sets of multiples, a subject which lies at the intersection of analytic and probabilistic number theory, has seen much development in recent years. The author gives a coherent, essentially self-contained account of the existing theory and at the same time brings the reader to the frontiers of research.

This book provides an introduction to quadratic forms that builds from basics up to recent results covering many aspects including lattice theory, Siegel's formula and results involving tensor products of positive definite quadratic forms. It is ideal for graduate students and researchers wishing for an insight into quadratic forms.

An introduction to the analytic theory of automorphic forms, limited to the case of fuchsian groups, but from the point of view of the general theory, ending with an introduction to unitary infinite dimensional representations. The main prerequisites are familiarity with functional analysis and elementary Lie theory.

The main theme of this book is the interplay between the behaviour of a class of stochastic processes (random walks) and discrete structure theory. Besides a detailed exposition of probabilistic and structure theoretic aspects, links with spectral theory and discrete potential theory are also discussed.

This book provides the first comprehensive introduction to the circle of ideas developed around Mori's program, the prerequisites being only a basic knowledge of the algebraic geometry. It will be of great interest to graduate students and researchers working in algebraic geometry and related fields.

Professor Motohashi shows that the Riemann zeta function is closely bound with automorphic forms and that many results from there can be woven with techniques and ideas from analytic number theory to yield new insights into, and views of, the function itself.

In this 1987 text Professor Jech gives a unified treatment of the various forcing methods used in set theory, and presents their important applications. Product forcing, iterated forcing and proper forcing have proved powerful tools when studying the foundations of mathematics, for instance in consistency proofs.

Affine differential geometry has undergone a period of revival and rapid progress. This book is a self-contained and systematic account of the theory from a more contemporary viewpoint, and emphasizes its relationship to other areas of differential geometry.

The theory of 3-transposition groups has become an important part of finite simple group theory. This book contains the first complete published proof of Fischer's Theorem on the classification of these groups. Part I is accessible to graduate students in finite groups while Parts II and III are aimed at specialists.

Algebraic coding has in recent years been increasingly applied to the study of combinatorial designs. Designs and their Codes gives an account of many of these applications together with a thorough general introduction to both design theory and code theory n developing the relationship between the two areas.

This book provides a clear exposition of the flourishing field of fixed point theory. Starting from the basics of Banach's contraction theorem, most of the main results and techniques are developed: fixed point results are established for several classes of maps and the three main approaches to establishing continuation principles are presented.

This 2001 treatment of the classical problem of derivation and integration combines the techniques of analysis with those of geometric measure theory. The main applications are to the Gauss-Green and Stokes theorems. The many results are proved in detail, and include background material and precise references to well-known results.

The large and diffuse body of literature connected with sign-solvability is presented as a coherent whole for the first time in this book, displaying it as a beautiful interplay between combinatorics and linear algebra.

The Schur algebra is an algebraic system providing a link between the representation theory of the symmetric and general linear groups. The approach is combinatorial where possible, making the presentation accessible to graduate students. This is the first comprehensive text in this important and active area of research; it will be of interest to all research workers in representation theory.

This advanced introduction emphasizes the variety of ideas, techniques, and applications of the subject. It discusses inequalities in Euclidean and Riemannian geometry, methods of classical differential geometry and elementary modern geometric measure theory, discretization of smooth spaces, and the influence of isoperimetric inequalities on heat diffusion on Riemannian manifolds.

The geometry and analysis that is discussed in this book extends to classical results for general discrete or Lie groups, and the methods used are analytical but have little to do with what is described these days as real analysis. Most of the results described in this book have a dual formulation.

In this tract Dr Ruston presents analogues for operators on Banach spaces of Fredholm's solution of integral equations of the second kind. Every effort has been made to keep the presentation as elementary as possible, using arguments that do not require a very advanced background, for those working in the general area of functional analysis and its applications.

Since the 1930s ergodic theory has been central to pure mathematics. This introduction provides sections on the classical ergodic theorems, topological dynamics, uniform distribution, Martingales, information theory and entropy. There is a chapter on mixing and one on special examples.

This tract presents an exposition of methods for testing sets of special functions for completeness and basis properties, mostly in L2 and L2 spaces. The emphasis on methods of testing and their applications will also interest scientists and engineers engaged in fields such as the sampling theory of signals in electrical engineering and boundary value problems in mathematical physics.

The book provides an introduction to the theory of cluster sets, a branch of topological analysis which has made great strides in recent years. An important and novel feature of the book is the discussion of more general applications to non-analytic functions, including arbitrary functions.

The representation theory of symmetric groups is one of the most beautiful, popular and important parts of algebra, with many deep relations to other areas of mathematics. Kleshchev describes a new approach to the subject, based on the recent work of Lascoux, Leclerc, Thibon, Ariki, Grojnowski and Brundan, as well as his own

This book presents a unified account of continuum percolation, a spatial random process that can be used to model many natural phenomena. The treatment is self-contained, assuming only familiarity with measure theory and basic probability theory. It will appeal to students and researchers in probability and stochastic geometry.

A self-contained introduction to analysis on fractals, a developing area of mathematics which focuses on the dynamical aspects of fractals. Suitable for graduate students and researchers in analysis and probability theory. Also useful as a supplementary text for graduate courses covering fractals, for both mathematicians and physicists.